"EarthBound Zero" and EarthBound's story are connected, but not nearly enough that you should play one to be able to understand anything that goes on in the other. The three games, though in the same series, were mostly designed so that they could each be picked up and played.
For what it's worth, EarthBound Zero and EarthBound share one character, do not occur in the same places, and little is referenced from the first game to its sequel. Additionally, remember that EarthBound was released in North America, while EarthBound Zero wasn't, which goes to show that the first game wasn't necessary at all.
Now then, as far as if you should play EarthBound Zero, that's up to you. EarthBound Zero is an NES RPG. It's difficult, requires grinding, luck (when you level, your bonuses are decided by the RNG), among other things. Where to go and what to do isn't always clearly defined, though the story eventually is defined. It's a great game, yes, but if you ask me, I'd really have to say that the game wasn't translated as well as it should have been, and that it's got several quirks that old RPGs do. By comparison, EarthBound is much more fun to play.
I suppose, then, my recommendation is to play EarthBound Zero if you really enjoy EarthBound and you want some more after you play it. I think it deserves to be remade, honestly, in an engine a little closer to its sequels, but that's just me. In fact, there are fans out there who are working on creating Earthbound Zero in EarthBound's engine, so you may even want to wait how ever many years it would take for that to happen.
As far as that site goes, I'm not all that sure if what he's doing is legal. Nintendo would certainly say it isn't, but essentially, he's selling roms of games he doesn't own the rights to. The legalities of free distribution of some of those games is disputed, yes, but selling them, however, is not. I just figured I'd mention that.
Edit: And Ian, I actually did play EarthBound way back when. I discovered Final Fantasy III, Chrono Trigger, and EarthBound all about the same time from rental stores before the N64 came out. In the end, I was able to obtain used copies of all three when those stores sold their old stock, and I even was able to get a copy of the player's guide that came with EB. I was pretty happy about that, I must say. Of course, I'll be honest, and admit that something terrible ruined the Chrono Trigger cartridge, which really saddens me
